8o On the PRINCIPLES of the 



ed, and that it is equally geometrical with the method of ex- 

 haullions of the ancients, who never fuppofed lines, furfaces, 

 or fblids, to be refolved into indefinitely fmall or infinitely little 

 elements. The expreflion infinitely little magnitude indeed im- 

 plies a contradidlion. For what has magnitude cannot be infi- 

 nitely little. 



This geometrical calculus, though it has no connection with 

 the various modifications of motion, is equally convenient in its 

 application with the method of fluxions, (which is unqueftiona- 

 bly a branch of general arithmetical proportion, in which i or 

 unit is the common ftandard of comparifon, as well as the con- 

 fequent of every ratio compounded, or decompounded). 



EXAMPLE I. 



In the circle ATB, (Fig. I. PI. I.) let the diameter AB be re- 

 prefented by D, TE perpendicular to it by Y, and AE by X. 

 Then (13. E. 6.) Y^ is equal to the retflangle DX — X\ But the 



antecedental of Y' is 2YY, and that of DX-X' is DX— 2XX, 

 (p. 6. Antecedental Calculus). Wherefore D — 2X is to 2Y as 



a a 



Y to X, that is, as TE to CE, (p. 9. Ant. Cat.). Confequently 

 CE is a third proportional to EO and TE. 



EXAMPLE n. 



To find the furface of the fphere of which ATBA is a great 

 circle, (Fig. L PI. 1.). 



The furface of the fpherical fegment, cut off by the circle, 

 of which TE is the radius, has to the fquare on any given line 

 B, a ratio compounded of the circumference of faid circle to B, 

 and of the antecedental of the curve AT to B, [Ant, CaL p. 9.) 

 But the antecedental of the curve is a fourth proportional to 2YD 



and 



